The Critical Rayleigh Number in Horizontal Convection for Pr = 1

We report the numerical simulations of the horizontal convection within a rectangle cavity tank at high Rayleigh numbers. The physical solution of horizontal convection depends on the spatial resolution of the meshes. The necessary mesh number N is proportion to Ra. The unstable numerical solutions are obtained asN < cRa. This power law also implies that the spatial resolution is dominated by viscosity and thermal diffusivity other than the length of the tank. Moreover, there is a Hopf bifurcation from steady solutions to unsteady solutions and the critical Rayleigh number Rac is obtained as 5.53 × 10 8 < Rac < 5.54 × 10 , which is much larger than the formerly obtained value. Horizontal convection, in which the water is unevenly heated at the horizontal surface, was taken as a model of abyssal ocean circulation. Unlike the Rayleigh-Bénard convection, the horizontal convection can be set to motion by any small temperature gradient. Moreover, the horizontal convection yields 1/5-power laws of Ra (e.g. Rossby, 1965; Siggers et al., 2004; Wang and Huang, 2005), comparing with the 1/4-power laws in the Rayleigh-Bénard convection. Similar to Rayleigh-Bénard convection, the horizontal convection may be unsteady at high Rayleigh numbers. There is a critical Rayleigh number Rac. The steady flow is unstable and become unsteady when Ra > Rac. The unsteady flow in horizontal convection was first found by numerical simulation Corresponding author email: [email protected]; [email protected].